I can only speak to the KM 1100 and say that regardless of weight, you won't get 4 corner registration. You can work to get 2-3 corners registered almost dead-on front-to-back, but one corner will always be off. The 8000s (previous model) were the same way. The technicians and "specialists" have told us as much. We have 2 of them and have to constantly re-register trays/stocks on one of them because the numbers used the previous day are no longer good the next day. That's only one of them and it isn't necessarily every single day, but frequently enough that our pressmen want registration sensitive jobs to go to the other 1100. And all of that will be worse with heavier weights (we run 111# cover fairly regularly). In multiple ways anything above 100# Cover (270 GSM) will be problematic.
Now the disclaimer, as per any machine/service experience comment: We're in Denver - so altitude/humidity probably play a part, as does the the service team involved, so your results may vary. And if you ONLY run 350 GSM stocks you may be able tune your press to handle that better (at the cost of lighter stocks being more of a problem). We run the gamut every single day so we can't really fine tune the press to any specific stock.